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Summary

At the end I want to summarize some findings of the previous six posts on Gossembrot. In my opinion, two main aspects seem important.

New types of labyrinths

Gossembrot has created two labyrinths with unique courses of the pathway, and thus designed two new types of labyrinths. The five-arm labyrinth on fol. 51 r is an outstanding type of labyrinth. The one-arm labyrinth with nine circuits on fol. 53 r is one of the rarer non-alternating types of labyrinths. Furthermore, a third new type of a four-arm labyrinth is hidden in the drawing on fol. 53 v.

Gossembrot could also have been first in drawing the Schedel type labyrinth (fol. 51 v) or the scaled-up basic type (fol. 54 v). It is true, that the manuscript containing the Schedel type is dated somewhat earlier than the one by Gossembrot. However, the drawing in Schedel manuscript could also have been added later. The two earliest examples of the scaled-up basic type are dated from the 15 th century without further precision. Thus, they could also have been generated later than 1480. However, I think this is unlikely. Both examples (Hesselager and Sibbo) were desinged in the classical style – which is the style that best matches with the natural way of designing this type of labyrinth.

Approaches to mazes

Gossembrot was strongly involved with the difference between labyrinth and maze. This is well attested by the mazes he had derived from the labyrinths of the Schedel type (fol. 52 r and fol. 52 above) and, following an other approach, from the Chartres type (fol. 54 r). And also by the fact that Gossembrot took this complex labyrinth for his best labyrinth.

I think also that his rejected design on fol. 53 v is not a mistaken attempt to the five-arm labyrinth on fol. 51 r. But instead, it seems to me that this is a failed attempt to derive a maze from the five-arm labyrinth. This is particularly supported by the design of the main axis. This was amended in a similar way as the ones of the mazes (fol. 52 r and fol. 52 v above) Gossembrot had derived from the Schedel type labyrinth.

It was not until the 15 th century that the creation of mazes began. The first drawing of a maze by Giovanni Fontana dates from year 1420 (see literature below: p. 138, fig. 239). Gossembrot was one of the first to draw mazes. His mazes, however, are, even compared with some other ones by Fontana (literature, p. 238, fig. 240), still rudementary and are fully based on unicursal labyrinths.

Conclusion

Gossembrot undoubtedly has his great importance in the design of unicursal labyrinths. Even if he must have been very fascinated by the maze, such that he himself took a maze for his best labyrinth, his drawings still represent tentative approaches and attempts to mazes. Contrastingly, he has created awesome original designs with fundamental innovations in unicursal labyrinths.

Literature
Kern H. Through the Labyrinth – Designs and Meanings over 5,000 Years. Munich, London, New York: Prestel 2000.

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The Four Labyrinths with 4 Arms und 8 Circuits

Four drawings by Gossembrot show labyrinths with 4 arms and 8 circuits. Among these, two each are on a circular and rectangular layout. Figure 1 shows these four figures compared. Figures a (circular) and c (rectangular) have the same course of the pathway (=). This is also true in figures b (circular) and d (rectangular). The two circular figures (a, b) as well as the two rectangular (c, d) have different courses of the path (≠).

Figure 1. The Four Designs Compared

All four figures bear inscriptions in their centers.

Figure a (fol. 51 v): „Laborintus inducens et educens“ – labyrinth leading in and leading out


Figure b (fol. 52 r): „Laborintus tamen educens nunquam intus perveniens fines“ – labyrinth leading out but nowhere arriving at the center

Figure c (fol. 52 v below): „Ibi introis et exis“ – here you enter and exit.

Figure d (fol. 52 v above): „Der Irrgang clausus est et numquam introibis“ the maze is closed and nowhere you enter.

From this we can see, that Gossembrot was engaged with the difference between labyrinth and maze. Figure 2 shows, using the lower, rectangular images, that the design of the side-arms in all four images is the same (areas within blue frames). The figures on the right side only differ with respect to the design of the main axis from those on the left side (areas within red frames). This becomes also clear from the patterns shown at the bottom of fig. 2. The left figures are labyrinths, the right figures are a special form of a simple maze. The pathway enters on the 6th circuit and there it branches. One branch continues to the first side-arm. There it turns to the 7th circuit, makes a full circuit and thereby traverses the main axis. It again turns at the first side-arm, leads back through the outer circuits 6 – 1 and arrives back in the other branch of the bifurcation. The innermost 8th circuit is completely isolated from the rest of the course of the pathway. It begins in a dead-end, does one round and ends in the center.

Figure 2. Labyrinth and Maze

So it seems, Gossembrot had derived a maze from the labyrinth. As a matter of fact, there exists a second historical labyrinth with the same pattern. This is sourced in a autograph (1456/63) of the Nuremberg physician and humanist Hartmann Schedel (see literature, below). The labyrinth drawn freehand was affixed to one of the last blank pages of the autograph. This autograph is accessible online in the same digital library as the manuscript by Gossembrot (further links, below). The original drawing of the labyrinth is oriented with the entrance to the left side. In fig. 3, for a better comparability, I have rotated it with the entrance to the bottom.

Figure 3. Type Schedel

Based on the earlier date (1456/63) of the publication by Schedel, I have named this type of labyrinth with „type Schedel“. Gossembrot was friends with Hermann Schedel, the uncle of Hartmann. The manuscript by Gossembrot dates from 1480. Having stated this, it has also to be considered that the labyrinth drawing of the Schedel autograph was affixed. Therefore it could also have been added later. Thus, it is well concievable that the drawings by Gossembrot were earlier and thus Gossembrot could have been the originator of this type of labyrinth.

Literature

  • Kern H. Through the Labyrinth: Designs and Meanings over 5000 years. London: Prestel 2000, p. 126, fig. 216.

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Further links

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There are 42 different one-arm alternating labyrinths with 7 circuits. Among these there is one pair of complementary interesting labyrinths. Now how does it look in pairs of complementary unintersting labyrinths? This question has already been indirectly answered in my last post (see related posts below): There is none! This sounds surprising. Therefore I address it further here. The 42 labyrinths form 21 complementary pairs. One of it is composed of 2 interesting labyrinths. We also know there are 22 interesting labyrinths. So the other 20 pairs are made up of an interesting and an uninteresting labyrinth each. Therefore no possibility remains for a pair with two complementary uninteresting labyrinths. What is the reason for that?

As we have seen, only in alternating labyrinths with an odd number of circuits it is possible to derive a complementary (see related posts). In such labyrinths the pathway always enters on an odd-numbered ciruit and also reaches the center from an odd-numbered circuit. Further, in one-arm labyrinths the pathway cannot enter the labyrinth on the same circuit from which it reaches the center.

In uninteresting labyrinths the pathway always must enter the labyrinth on the outermost circuit or reach the center from the innermost circuit. The complementary is derived by mirroring. By this, the outermost is transformed to the innermost circuit and vice versa. If in an original labyrinth the pathway enters on the first circuit, it is an uninteresting labyrinth. In the complementary the path will enter on the innermost circuit. Thus the complementary is not an uninteresting labyrinth, unless the path would reach the center from the innermost circuit. This, however is not possible, as it already enters the labyrinth on this circuit. The original is an unintersting, but the complementary an interesting labyrinth. The other alternative would be that the path in the original labyrinth reached the center from the innermost circuit. But then in the complementary it would reach the center from the outermost circuit what is not an unintersting labyrinth. Therefore the complementary could only be an unintersting labyrinth, if the path would enter it on the outermost circuit. This, however is impossible, as the path reaches the center from this circuit.

These results are only valid for one-arm labyrinths with up to 7 circuits. In labyrinths with mulitiple arms, the pathway may reach the center from the same circuit on which it enters the labyrinth. Thus, for example it could enter the original labyrinth on the first circuit and also reach the center from the first circuit. This would consitute an uninteresting labyrinth. In the complementary, the pathway would then enter the labyrinth on the innermost circuit and also reach the center from the innermost circuit, what again would qualify for an uninteresting labyrinth. In one-arm labyriths with more thean 7 circuits the definition of what constitutes an uninteresting labyrinth can be extended. In these cases trivial circuits can be added not only at the outside or inside of smaller interesting labyrinths (what generates uninteresting labyrints) but also on central circuits between other interesting elements at the inside and outside of the labyrinth, what also may generate uninteresting labyrinths.

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I have already elaborated on uninteresting and interesting labyrinths (see related posts, below). Unintersting labyrinths can be generated by simply attaching additional trivial circuits to the outside or inside of smaller labyrinths. Interesting labyrinths cannot be obtained this way. This particularly implies, that in interesting labyrinths the pathway may not enter the labyrinth on the outermost circuit or reach the center from the innermost circuit. The dual of an interesting labyrinth is another interesting labyrinth too.

This is different if we derive the complementary labyrinth of an interesting labyrinth. The resultung labyrinth may very well be an uninteresting labyrinth. Complementary labyrinths exist only for alternating labyrinths with an odd number of circuits. To obtain the complementary, the pattern of the original labyrinth is vertically mirrored without interrupting the connections between the outside or center of the labyrinth with their corresponding circuits. Labyrinths with an odd number of circuits always have a central circuit. When the pattern is mirrored, this circuit remains in position, whilst the other circuits change their positions symmetrically around it.

Figure 1. Mirroring

In a labyrinth with seven circuits, e.g., the central circuit is the one with number 4. After the mirroring, this remains in its position as number 4. The outermost circuit, number 1, transforms to the innermost circuit and obtains number 7, circuit 2 changes to circuit 6, circuit 3 to circuit 5, and vice versa.

Now, if in an interesting labyrinth the pathway first leads to the innermost circuit or reaches the center from the outermost circuit, then the complementary to this labyrinth will be an uninteresting labyrinth. This, because in the complementary, the path will enter the labyrinth on the outermost or reach the center from the innermost circuit. Thus, there exist pairs of complementary labyrinths, both of which are interesting and others in which one of the labyrinths is interesting and the other uninteresting.

Now I want to find out which are the pairs of interesting complementary labyrinths. The website of Tony Phillips provides best material for such a purpose. On one page, HERE, are included the seed patterns (left figures) and the patterns (right figures) of the interesting alternating labyrinths with up to 7 circuits. I therefore reproduce the page in fig. 2 and in the following add some comments to the items indicated with red letters:

Figure 2. Interesting Labyrinths

  • a) In addition to the circuits, Tony also counts the exterior (= 0) and the center (= one greater than the number of circuits) of a labyrinth. He refers to this as the depth of the labyrinth. A labyrinth with depth 4, thus, has three circuits, one with depth 6 has 5 circuits and so on.
  • b) Below the two figures (seed pattern und pattern), in each case the sequence of circuits is listed. This also contains the zero for the exterior and the number for the center, here indicated with red boxes. The true sequence of circuits is the sequence of numbers between these boxes.
  • c) If the labyrinth is self-dual, this is indicated as „s.d.“ after the sequence of circuits.
  • d) If this is not the case, anyway only one of each dual example is shown in the figures. However, the sequence of circuits of the dual not shown is listed in parentheses below the sequence of circuits of the labyrinth shown.
  • e) The patterns are drawn in such a manner that the course of the pathway leads from top right to bottom left. This is different from how I do it. I draw the pattern from top left to bottom right. As a consequence, the labyrinth that corresponds with the pattern by Tony rotates anti-clockwise, whereas in my case it rotates clockwise.
  • f) Now, lets consider all interesting (including very interesting) labyrinths with 7 circuits. Of these, there are 22 (6 of them very interesting) interesting labyrinths. In fig. 2 the seed patterns and patterns of only 14 labyrinths are depicted. The missing 8, however, are duals, represented by the sequences of circuits in parentheses.

Among the interesting labyrinths with 7 circuits, only 2 exist, in which the pathway does not enter the labyrinth on the innermost circuit nor reach the center from the outermost circuit. And these two form the only pair of interesting labyrinths complementary to each other. We already know this pair from the first post of this series. It is the basic type labyrinth (g) and the labyrinth with the S-shaped course of the pathway (h).

Figure 3. Complementary and Interesting Labyrinths

These are self-dual and thus very interesting labyrinths. In the other 20 interesting labyrinths, the complementary in each case is an uninteresting labyrinth.

Thus, there are 42 different alternating labyrinths with one arm and 7 circuits. Among these, there are 8 pairs of interesting dual labyrinths, 6 self-dual very interesting labyrinths, but only 1 pair of interesting complementary labyrinths. In addition, there is no pair of interesting complementary labyrinths with less than 7 circuits.

Pairs of complementary interesting labyrinths seem to be relatively rare and thus something special.

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It is known that there are 8 alternating labyrinths with 1 arm and 5 circuits (see “Considering Meanders and Labyrinths”, related posts, below). Of these, four are not self-dual. These four all are in a relationship to each other via the duality and complementarity (see “The Complementary versus the Dual Labyrinth”, related posts, below). The other four labyrinths are self-dual.

I had already pointed to the relationship between complementary and self-dual labyrinths (see “The Complementary Labyrinth”, related posts, below). Here I want to elaborate on it further. For this purpose I use the same form of diagram I had already used in my previous post (see “The Complementary versus the Dual Labyrinth”). I also use the same numbers of the labyrinths according to the numbering of Arnol’d’s meanders (see “Considering Meanders and Labyrinths”), that underlie them.

Figure 1. Labyrinths 1 and 6

The first of the Arnol’d’s labyrinths, number 1, is self-dual. In the diagram, the dual is situated in the same row, the complementary in the same column with the original labyrinth. The dual of number 1 is again number 1 (what actually is the meaning of selfdual). The complementary of number 1 is number 6. And – of course – is the dual to the complementary again number 6. So in the case of self-dual labyrinths, we only captured two different labyrinths, whereas it were four in the case of not self-dual labyrinths.

Thus, two more labyrinths are still missing. We need another diagram to capture labyrinths number 3 and number 8 (fig. 2).

Figure 2. Labyrinths 3 and 8

And, indeed, these two are complementary to each other. So in self-dual labyrinths, only two different labyrinths are in a relationship to each other.

Here the question arises: Do there also exist self-complementary labyrinths? Up to now we have not yet found such a labyrinth. So let us remember, what self-dual imples. The patterns of the original and self-dual labyrinths are self-covering. In fig. 3 I show what that means. The two patterns in the same row are dual. If we shift them together, we can easily see, what I mean.

Figure 3. Self-dual patterns are self-covering

Thus, self-complementary would imply that the original and complementary pattern would also be self-covering.

Figure 4. Complementary patterns are not self-covering

Fig. 4 shows, that even though there is a certain similarity between these two patterns, they are not self-covering. In my opinion there are no self-complementary labyrinths. This is because vertical mirroring with uninterrupted connections to the entrance and center modifies the sequence of circuits. This, however, woult have to remain unaltered.

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In the last post I have presented the complementary labyrinth. I did this with the example of the basic type labyrinth. This is a self-dual labyrinth. The complementary is different from the dual labyrinth. This can be better shown using non-self-dual labyrinths. I want tho show this here and for this choose an alternating labyrinth with 1 arm and 5 circuits. As already shown in this blog, there exist 8 such labyrinths (see related post below: Considerung Meanders and Labyrinths). Of these, 4 are self-dual (labyrinths 1, 3, 6, and 8) and 4 are not self-dual (labyrinths 2, 4, 5, and 7).

I thus choose one of the non-self-dual labyrinths, nr. 2, and use the pattern of it. With the pattern, two activities can be performed:

  • Rotate

  • Mirror

Figure 1 shows the result of performing these actions with pattern 2.

Figure 1. Rotating and Mirroring of the Pattern

Rotation leads to the pattern of labyrinth 4
Mirroring leads to pattern 7

So we have already three labyrinths. Now it is possible to go even further. Rotating the dual again brings it back to the original labyrinth. However, the dual can also be mirrored. This results then in the complementary of the dual. And similarly, the complementary can be rotated, which results in the dual to the complementary.

Mirroring of the dual (pattern 4) leads to the complementary pattern of labyrinth 5
Rotation of the complementary (pattern 7) leads to the dual of it – which is also pattern 5.

Figure 2. Relationships

Figure 2 shows the labyrinths corresponding to the patterns. The labyrinths are presented in basic form (i.e shown with their walls delimiting the pathway) in the concentric style. All four non-self-dual alternating labyrinths with 1 arm and 5 circuits are in a relation of either dualtiy or complementarity to each other.

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An interesting labyrinth is reproduced in the book of Kern (fig. 200, p. 119)°. A drawing by Arabian geographer Al Qazvini in his cosmography completed in 1276 is meant to show the ground plan of the residence of the ruler of Byzantium, before the large city of Constantinople was built up.

This non-alternating labyrinth has 10 circuits and a unique course of the pathway. I will show this using the Ariadne’s Thread and the pattern. In my post “From the Ariadne’s Thread to the Pattern – Method 2” (see related posts, below), I have already described how the pattern can be obtained. When deriving the pattern I always start with a labyrinth that rotates clockwise and lies with the entrance from below. The labyrinth by Qazvini rotates in clockwise direction, however it lies with the entrance from above. Therefore I rotate the following images of the labyrinth by a semicircle so that the entrance comes to lie from below. So it is possible to follow the course of the pathway with the Ariadne’s Thread and in parallel see how this is represented in the pattern.

Four steps can be distinguished in the course of the pathway.

Phase 1

The path first leads to the 3rd circuit. The entrance is marked with an arrow pointing inwards. In the pattern, axial sections of the path are represented by vertical, circuits by horizontal lines. The way from the outside in is represented from above to below.

Phase 2

In a second step, the path now winds itself inwards in the shape of a serpentine until it reaches the 10th and innermost circuit. Up to this point the course is alternating.

Phase 3

Next follows the section where the pathway leads from the innermost to the outermost circuit whilst it traverses the axis. In order to derive the pattern, the labyrinth is split along the axis and then uncurled on both sides. As the pathway traverses the axis, the piece of it along the axis has to be split in two halves (see related posts below: “The Pattern in Non-alternating Labyrinths”). This is indicated with the dashed lines. These show one and the same piece of the pathway. In the pattern, as all other axial pieces, this is represented vertically, however with lines showing up on both sides of the rectangular form and a course similarly on both sides from bottom to top.

Phase 4

Finally the pathway continues on the outermost circuit in the same direction it had previously taken on the innermost circuit (anti clockwise), then turns to the second circuit, from where it reaches the center (highlighted with a bullet point).

Related Posts:

°Kern, Hermann. Through the Labyrinth – Designs and Meanings over 5000 Years. Munich: Prestel, 2000.

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